Integrable pairing models in

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Integrable pairing models in cold atom
physics The Richardson-Gaudin models

Jorge Dukelsky. IEM. CSIC.

• Cooper pairs and BCS (1956-1957)
• Richardson exact solution (1963).
• Ultrasmall superconducting grains (1999).
• SU(2) Richardson-Gaudin models (2000).
• Cooper pairs and pairing correlations from the
  exact solution. BCS-BEC crossover in cold atoms
  (2005).
• Generalized Richardson-Gaudin Models for rgt1
  (2006-2009). 3-color pairing. T0,1 p-n pairing
  model and spin 3/2 atoms.
• p-wave integrable pairing Hamiltonian (2009).

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The Cooper Problem
Problem A pair of electrons with an attractive
interaction on top of an inert Fermi sea.
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Bound pair for arbitrary small attractive
interaction. The FS is unstable against the
formation of these pairs
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Richardsons Exact Solution
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Exact Solution of the BCS Model
Eigenvalue equation Ansatz for the eigenstates
(generalized Cooper ansatz)
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• Richardson equations

• Properties
• This is a set of M nonlinear coupled equations
  with M unknowns (E?).
• The first and second terms correspond to the
  equations for the one pair system. The third term
  contains the many body correlations and the
  exchange symmetry.
• The pair energies are either real or complex
  conjugated pairs.
• There are as many independent solutions as states
  in the Hilbert space.
• The solutions can be classified in the weak
  coupling limit (g?0).

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Pair energies E for a system of 200 equidistant
levels at half filling
In the thermodynamic limit Richardson ? BCS an
equation for the arc in the complex plane.
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Condensation energy for even and odd grains PBCS
versus Exact J. Dukelsky and G. Sierra, PRL 83,
172 (1999)
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The Structure of the Cooper pairs in BCS-BEC
rs Interparticle distance , ? Size of the
Cooper pair
Quasibound molecules Pair resonaces
Free fermions
Quasibound molecules Pair
resonances
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What is a Cooper pair in the superfluid is
medium? G. Ortiz and J. Dukelsky, Phys. Rev. A
72, 043611 (2005)
Cooper pair wavefunction

• From MF BCS
• From pair correlations
• From Exact wavefuction

• E real and lt0, bound eigenstate of a zero range
  interaction parametrized by a.
• E complex and R (E) lt 0, quasibound molecule.
• E complex and R (E) gt 0, molecular resonance.
• E Real and gt0 free two particle state.

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BCS-BEC Crossover diagram
f pairs with Re(E) gt0 1-f unpaired, E real gt0
f1 Re(E)lt0
?1/kf as

• -1, f 0.35 (BCS)
• 0, f 0.87 (BCS)
• 0.37, f 1 (BCS-P)
• 0.55, f 1 (P-BEC)
• 1,2, f1 (BEC)

f1 some Re(E)gt0 others Re(E) lt0
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Cooper pair wave function
Weak coupling BCS
Strong coupling BCS
BEC
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Sizes and Fraction of the condensate
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Nature 454, 739-743 (2008)
Cooper wavefunction in the BEC region
A spectroscopic pair size can be defined from the
threshold energy of the pair dissociation
spectrum as
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The Hyperbolic Richardson-Gaudin Model
A particular RG realization of the hyperbolic
family is the separable pairing Hamiltonian
With eigenstates
Richardson equations
The physics of the model is encoded in the exact
solution. It does not depend on any particular
representation of the Lie algebra
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(pxipy) SU(2) spinless fermion representation
Choosing ?k k2 we arrive to the pxipy
Hamiltonian
M. I. Ibañez, J. Links, G. Sierra and S. Y. Zhao,
Phys. Rev. B 79, 180501 (2009). C. Dunning, M. I.
Ibañez, J. Links, G. Sierra and S. Y. Zhao,, J.
Stat. Mech. P080025 (2010). S. Rombouts, J.
Dukelsky and G. Ortiz, ArXiv1008.3406.

• It is know that p-wave pairing has a QPT
  separating two gapped phases
• A non-trivial topological phase. Weak pairing.
• A phase characterized by tightly bound
  quasi-molecules. Strong pairing.
• N. Read and D. Green, Phys. Rev. B 61, 10267
  (2000).
• Moreover, there is a particular state ( the
  Moore-Read Pfafian) isomorphic to the a
  fractional quantum Hall GS.

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The presence of a zero energy level with variable
degeneracy determines the physics of the model
From the Richardson equations the necessary
condition to have N pairons converging to zero,
Ea -gt 0, is
1) No pairons converge to zero
QPT
2) All pairons converge to zero (Moore-Read line)
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Quantum phase diagram of the hyperbolic
model
pxipy
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Pairons distribution in a Disk of R18 with total
degeneracy L504 and M126. (quarter
filling) g0.5 weak coupling g1.5 weak
pairing g2.5 strong pairing
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(No Transcript)
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Momentum density profiles for L504 and M 126.
Exact versus BCS
BCS
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Higher order derivatives of the GS energy in the
thermodynamic limit
Possible third-order phase transition in the
large-N lattice gauge theory D. J. Gross and E.
Witten, Phys. Rev. D 21, 446453 (1980)
3º order QPT
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Characterization of the QPT
In the thermodynamic limit the condensate
wavefunction in k-space is
The length scale can be calculated as
Accessible experimentally by quantum noise
interferometry and time of flight analysis?
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A similar analysis can be applied to the pairs in
the exact solution
The root mean square of the pair
wavefunction is finite for E complex or real and
negative. However,
for In strong pairing all pairs have finite
radius. At the QPT one pairon becomes real an
positive corresponding to a single deconfined
Cooper pair on top of an ensemble of quasi-bound
molecules.
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Exactly Solvable Pairing Hamiltonians
1) SU(2), Rank 1 algebra
2) SO(5), Rank 2 algebra
J. Dukelsky, V. G. Gueorguiev, P. Van Isacker, S.
Dimitrova, B. Errea y S. Lerma H. PRL 96 (2006)
072503.
3) SO(6), Rank 3 algebra
B. Errea, J. Dukelsky and G. Ortiz, PRA 79 05160
(2009)
4) SO(8), Rank 4 algebra
S. Lerma H., B. Errea, J. Dukelsky and W.
Satula. PRL 99, 032501 (2007).
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3-color Pairing
Phase diagram
Breached, Unbreached configurations
L500, N150, P(NG-NB)/(NGNB)
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3-color Pairing
Density profiles
Occupation probabilities
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Summary

• For finite system, the exact solution
  incorporates mesoscopic correlations absent in
  BCS and PBCS.
• From the analys is of the exact Richardson
  wavefunction we proposed a new view to the nature
  of the Cooper pairs in the BCS-BEC transition for
  s-wave and p-wave pairing.
• The hyperbolic RG offers a unique tool to study
  a rare 3º order QPT for p-wave pairing.
• The root mean square size of the pair wave
  function diverges at the critical point. It could
  be a clear experimental signature of the QPT.
• Extensions to higher rank algebras include
• The SO(6) model for three-component systems
  describing color superconductivity and exotic
  phases with two condensates .
• The SO(8) model for four-components systems (
  3/2 fermions or T0,1 proton-neutron pairing) .
  Competence between pair and quartet correlations.
  

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• Construction of the Integrals of Motion
• The most general combination of linear and
  quadratic generators, with the restriction of
  being hermitian and number conserving, is

• The integrability condition
  leads to

• These are the same conditions encountered by
  Gaudin (J. de Phys. 37 (1976) 1087) in a spin
  model known as the Gaudin magnet.

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• Gaudin (1976) found three solutions

XXX (Rational)
XXZ (Hyperbolic)

• Hamiltonianos

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Exactly Solvable RG models for simple Lie algebras
Cartan classification of Lie algebras
rank An su(n1) Bn so(2n1) Cn sp(2n) Dn so(2n)
1 su(2), su(1,1) pairing so(3)su(2) sp(2) su(2) so(2) u(1)
2 su(3) Three level Lipkins so(5), so(3,2) pn-pairing sp(4) so(5) so(4) su(2)xsu(2)
3 su(4) Wigner so(7) FDSM sp(6) FDSM so(6)su(4) color superconductivity
4 su(5) so(9) sp(8) so(8) pairing T0,1. Ginnocchio. 3/2 fermions